An Interactive Guide to the Fourier Transform

I have not read the whole article. But, what is shown at the beginning is not the Fourier Transform, it is the Discrete Fourier Transform (DFT).

Though the DFT can be implemented efficiently using the Fast Fourier Transform (FFT) algorithm, the DFT is far from being the best estimator for frequencies contained in a signal. Other estimators (like Maximum Likelihood [ML], [Root-]MUSIC, or ESPRIT) are in general far more accurate - at the cost of higher computational effort.

Not a particularly fair comparison, the DFT is a non-statistical operation.

Why do you think, that it is not fair?

You can even use these algorithms with a single snapshot (spatial smoothing).

Can you provide more details please?

The FFT is still easy to use, and it you want a higher frequency resolution (not higher max frequency), you can zero pad your signal and get higher frequency resolution.

Zero-padding gives you a smoother curve, i.e., more points to look at. But it does not add new peaks. So, if you have two very close frequencies that produce a single peak in the DFT (w/o zero-padding), you would not get two peaks after zero-padding. In the field, were I work, resolution is understood as the minimum distance between two frequencies such that you are able to detect them individually (and not as a single frequency).

Zero-padding helps you to find the true position (frequency) of a peak in the DFT-spectrum. So, your frequency estimates can get better. However, the peaks of a DFT are the summits of hills that are usually much wider than compared to other techniques (like Capon or MUSIC) whose spectra tend to have much narrower hills. Zero-padding does not increase the sharpness of these hills (does not make them narrower). Likewise the DFT tends to be more noisy in the frequency domain compared to other techniques which could lead to false detections (e.g. with a CFAR variant).